Problem 15 - Thinking Mathematically

Question

Select the statement that is equivalent to I saw the original King Kong or the 2005 version. a. If I did not see the original King Kong, I saw the 2005 version. b. I saw both the original King Kong and the 2005 version. c. If I saw the original King Kong, I did not see the 2005 version. d. If I saw the 2005 version, I did not see the original King Kong.

Step-by-Step Solution

Answer

The statement that is equivalent to the original statement 'I saw the original King Kong or the 2005 version' is option a: 'If I did not see the original King Kong, I saw the 2005 version.'

Step 1: Analysis of Option a

Look at option a: 'If I did not see the original King Kong, I saw the 2005 version.' This statement is the logical construct of an implication, where the second event (seeing the 2005 version) occurs if the first event (not seeing the original King Kong) is true. This mirrors the original statement because in both cases, at least one version of the movie is seen.

Step 2: Analysis of Options b, c and d

Now look at options b, c, and d. Option b: 'I saw both the original King Kong and the 2005 version' suggests both events occurred, which is a logical conjunction, not disjunction. It could be untrue even if the original statement is true. Therefore, it is not equivalent. For option c: 'If I saw the original King Kong, I did not see the 2005 version', this implies a logical construct that contradicts the original statement. Option c suggests that both events cannot occur simultaneously, whereas the original statement doesn't exclude this possibility. Option d: 'If I saw the 2005 version, I did not see the original King Kong' presents the same contradiction as option c. Therefore, options b, c, and d are not equivalent to the original statement.

Step 3: Selection of the Equivalent Statement

After analyzing all the options, it is clear that option a: 'If I did not see the original King Kong, I saw the 2005 version' is the only statement equivalent to the original statement: 'I saw the original King Kong or the 2005 version.' This is because it preserves the logical construct of a disjunction, where at least one of the two events must occur.

Key Concepts

Disjunction

A disjunction is a fundamental concept in logic that refers to the use of the word "or" to combine two statements. In the statement, "I saw the original King Kong or the 2005 version," the disjunction implies that at least one of these events must be true. This means you could have seen either one of the versions, or even both, for the statement to hold true.

Disjunctions are represented symbolically using the logical "or," expressed in logic notation as \( p \lor q \), where \( p \) and \( q \) are separate statements. A disjunction is true if at least one of the components \( p \) or \( q \) is true. This makes it a very flexible logical construct, allowing for multiple possible truths without exclusion.

Logical Implication

Logical implication is another key concept in logic. It involves a situation where the truth of one statement guarantees the truth of another. This is denoted with an "if...then..." structure. For example, "If I did not see the original King Kong, then I saw the 2005 version."

This statement can be expressed symbolically as \( eg p \rightarrow q \), which reads "if not \( p \), then \( q \)." Here, \( eg p \) signifies the negation of the first statement, and \( q \) stands for the second statement. A logical implication is true in all scenarios except when the first statement is true and the second is false.

• This means if the original movie wasn't seen, the implication dictates that the 2005 version must have been.
• This maintains logical consistency with a disjunction due to the inclusive nature of "or."

Equivalence Analysis

Analyzing logical equivalence involves identifying statements that reveal the same truth regardless of how they are expressed. In our problem, “I saw the original King Kong or the 2005 version" needs to be equivalent to the options provided.

In the analysis:

• Option a, "If I did not see the original King Kong, then I saw the 2005 version," accurately captures the essence of the original disjunction. This logical implication implies at least one version is seen, aligning with the original statement's meaning.
• Option b suggests both movies were seen, representing a conjunction, which is not equivalent because the original disjunction allows for seeing just one version.
• Options c and d imply exclusivity, which contradicts the original inclusive disjunction.

Logical equivalence requires identifying the transformation or reinterpretation of a statement that retains the original truth values under all circumstances.

Logical Constructs

Logical constructs are the foundational building blocks of logical reasoning and analysis. In the given exercise, the main constructs include disjunctions and implications.

• Disjunction allows for multiple possibilities, accommodating flexibility and inclusiveness in logical statements.
• Implication introduces a cause-effect relationship between statements, where one being true influences or necessitates the truth of another.

Understanding these logical constructs allows us to determine truth values across various permutations of statements. This is crucial when analyzing complex logical expressions or arguments in mathematics or computer science.

By mastering logical constructs, like those demonstrated in the problem, one becomes adept at breaking down and interpreting intricate information, which is a valuable skill across numerous disciplines.